Question
Answer
$$\dfrac{4}{4-4\sqrt{2}}=-(1+\sqrt{2})$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{4}{4-4\sqrt{2}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{4}{4-4\sqrt{2}}\frac{4+4\sqrt{2}}{4+4\sqrt{2}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{16+16\sqrt{2}}{16+16\sqrt{2}-16\sqrt{2}-32} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{16+16\sqrt{2}}{-16} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{1+\sqrt{2}}{-1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}-\frac{1+\sqrt{2}}{1} \xlongequal{ } \\[1 em] & \xlongequal{ }-(1+\sqrt{2})\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 4 + 4 \sqrt{2}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 4 } \cdot \left( 4 + 4 \sqrt{2}\right) = \color{blue}{4} \cdot4+\color{blue}{4} \cdot 4 \sqrt{2} = \\ = 16 + 16 \sqrt{2} $$ Simplify denominator. $$ \color{blue}{ \left( 4- 4 \sqrt{2}\right) } \cdot \left( 4 + 4 \sqrt{2}\right) = \color{blue}{4} \cdot4+\color{blue}{4} \cdot 4 \sqrt{2}\color{blue}{- 4 \sqrt{2}} \cdot4\color{blue}{- 4 \sqrt{2}} \cdot 4 \sqrt{2} = \\ = 16 + 16 \sqrt{2}- 16 \sqrt{2}-32 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 16. |
| ⑤ | Place a negative sign in front of a fraction. |
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